List of top Mathematics Questions asked in AP EAPCET

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that at least one of them is a face card is:

A person is known to speak the truth in 3 out of 4 occasions. If he throws a die and reports that it is six, then the probability that it is actually six is:

If a discrete random variable \(X\) has the probability distribution \[ P(X = x) = k \frac{2^{2x+1}}{(2x+1)!}, \quad x=0,1,2,\ldots, \] then find \(k\).

If \(\vec{a} = 2 \vec{i} - 3 \vec{j} + 4 \vec{k}, \vec{b} = \vec{i} + 2 \vec{j} - \vec{k}, \vec{c} = -3 \vec{i} - \vec{j} + 2 \vec{k}\) and \(\vec{d} = \vec{i} + \vec{j} + \vec{k}\) are four vectors, then evaluate \[ (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = ? \]

If \(\vec{a} = \vec{i} + p \vec{j} - 3 \vec{k}, \vec{b} = p \vec{i} - 3 \vec{j} + \vec{k}, \vec{c} = -3 \vec{i} + \vec{j} + 2 \vec{k}\) are three vectors such that \[ |\vec{a} \times \vec{b}| = |\vec{a} \times \vec{c}|, \] then \(p =\)

The variance of the ungrouped data \(2, 12, 3, 11, 5, 10, 6, 7\) is:

In \(\triangle ABC\), if \(r_1 = 2r_2 = 3r_3\), then find the ratio \(a : b\).

In \(\triangle ABC\), the sum of the lengths of two sides is \(x\) and the product of those lengths is \(y\). If \(c\) is the length of its third side and \(x^2 - c^2 = y\), then the circumradius of that triangle is:

If \(\vec{a}, \vec{b}, \vec{c}\) are three unit vectors such that \[ |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 15, \] then \[ |\vec{a} - \vec{b} - \vec{c}|^2 - 4(\vec{b} \cdot \vec{c}) = ? \]

Let \(\vec{a}\), \(\vec{b}\) be position vectors of points \(A\) and \(B\) respectively. \(C\) and \(D\) are points on the line \(AB\) such that \(\overrightarrow{AB}, \overrightarrow{AC}\) and \(\overrightarrow{BD}, \overrightarrow{BA}\) are two pairs of like vectors. If \(\overrightarrow{AC} = 3 \overrightarrow{AB}\) and \(\overrightarrow{BD} = 2 \overrightarrow{BA}\), then \(\overrightarrow{CD} =\)

Let \(\vec{2i} - \vec{j} - \vec{k}, \vec{5i} + \vec{j} - 2\vec{k}, -13\vec{i} - 11 \vec{j} + 4 \vec{k}\) be the position vectors of three points \(A, B, C\) respectively. If \(\overrightarrow{AB} = \lambda \overrightarrow{BC}\) and \(\overrightarrow{AC} = \mu \overrightarrow{CB}\), then \(\lambda + \mu =\)

If the area of triangle \(ABC\) is \(4\sqrt{5}\) sq. units, length of the side \(CA\) is 6 units and \(\tan \frac{B}{2} = \frac{\sqrt{5}}{4}\), then its smallest side is of length:

Let \( H(x) = 3x^4 + 6x^3 - 2x^2 + 1 \) and \( g(x) \) be a linear polynomial. If \[ \frac{H(x)}{(x-1)(x+1)(x-2)} = f(x) + \frac{g(x)}{(x-1)(x+1)(x-2)}, \] then find \( H(-1) + 2H(2) - 3H(1) \).

If \(630^\circ<\theta<810^\circ\) and \(\tan \theta = -\frac{7}{24}\), then find \(\cos \left(\frac{\theta}{4}\right)\).

For \(\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), if \(2\cos \theta + \sin \theta = 1\), and \(7\cos \theta + 6 \sin \theta = k\), then the possible values of \(k\) are:

Evaluate \[ \sum_{k=0}^{12} \sin\left( (k+1) \frac{\pi}{6} + \frac{\pi}{4} \right) \sin \left( \frac{k \pi}{6} + \frac{\pi}{4} \right) \]

Number of solutions of the equation \[ 2 \sin^2 \theta - 3 \cos^2 \theta = \sin \theta \cos \theta \] in the interval \((- \pi, \pi)\) is:

Evaluate \[ \tan^{-1} \frac{\sqrt{8} - 2\sqrt{15}}{\sqrt{15} + 1} + \tan^{-1} \frac{1}{\sqrt{5}} \]

If \(x\) is so large that terms containing \(x^{-3}\), \(x^{-4}\), \(x^{-5}\), \ldots can be neglected, then the approximate value of \[ \left(\frac{3x - 5}{4x^2 + 3}\right)^{-4/5} \] is:

The number of all possible positive integral solutions of the equation \(xyz = 30\) is:

The number of ways of arranging all the letters of the word PERFECTION such that there must be exactly two consonants between any two vowels is:

If \(\alpha\) and \(\beta\) (\(\alpha>\beta\)) are the multiple roots of the equation \[ 4x^4 + 4x^3 - 23x^2 - 12x + 36 = 0, \] then find \(2\alpha - \beta\).

If \( z_1 \) and \( z_2 \) are two of the \( n^{th} \) roots of unity such that the line segment joining them subtends a right angle at the origin, then for a positive integer \( k \), \( n \) takes the form:

\[ \left( \sqrt{2} + 1 + i \sqrt{2} - 1 \right)^8 = ? \]

If the harmonic mean of the roots of the equation \[ \sqrt{2} x^2 - b x + \left( 8 - 2\sqrt{5} \right) = 0 \] is 4, then the value of \( b \) is: